Quadric Surfaces — Question 7

Question 7 – Solution

Strategy Because $z$ is absent, any planar curve in $xy$ is extruded parallel to the $z$-axis.

Standard form Completing squares yields $$9(x-1)^2+4(y+2)^2=36, \quad\boxed{\frac{(x-1)^2}{4}+\frac{(y+2)^2}{9}=1}.$$ This is an elliptic cylinder parallel to the $z$-axis.

Cross-section Its semiaxes are 2 and 3, so each perpendicular cross-section has area $\boxed{6\pi}$.

$z$-axis test At $x=y=0$, the left side of the original equation is $-11\ne 0$, so the cylinder does not meet the $z$-axis.